The group \(PSL_{3}(3)\)  has two conjugacy classes of subgroups of order 3,
say \(\mathcal {K}_{1}\)  and \(\mathcal {K}_{2}\)  by [1]}. If \(H_{1}\in \mathcal {K}_{1}\) ,
then \(H_{1}\)  is a group of elations of \(PG_{2}(3)\)  having the same center \(C\) 
and the same axis \(\ell \)  (e.g. see [2]}). Hence, \(H_{1}\)  fixes seven
flags of \(PG_{2}(3)\) . If \(H_{2}\in \mathcal {K}_{2}\) , then \(H_{2}\)  normalizes
six suitable Sylow 13-subgroups of \(PSL_{3}(3)\) , and fixes a unique flag
of \(PG_{2}(3)\) . No further points or lines of \(PG_{2}(3)\)  are fixed by \(H_{2}\) . Finally, \(\sigma \)  is a symmetric polarity of \(PG_{2}(3)\) . Thus \(\sigma \) 
switches points and lines of \(PG_{2}(3)\)  and hence preserves both \(\mathcal {K}_{1}\)  and \(\mathcal {K}_{2}\) . Therefore, \(\mathcal {K}_{1}\)  and \(\mathcal {K}_{2}\)  are also conjugacy \(PSL_{3}(3):\left\langle \sigma \right\rangle \) -classes.
